Wednesday Math Colloquium Talks
What factors
influence achievement in remedial mathematics classes?
This study examined the
predictive qualities that cognitive and motivational variables have on remedial
mathematics students at an urban four-year university in southern
A Unified View of
Number-theoretic Sums
Two types of sums of arithmetic
functions seem to dominate the landscape of number theory, namely, the divisor sums
and partial sums. Our first order of business is to generalize the classical
definitions of arithmetic functions, divisor sums and partial sums. Indeed, by
selecting suitable entry points to these two types of sums, an interesting
kinship emerges between them and they seem to share a highly common genealogy.
We will continue to explore the power of iteration initiated in our
Student Attitudes,
Conceptions, and Achievement in Introductory
The purpose of this study was to measure student attitudes and conceptions, as well as misconceptions, in introductory undergraduate college statistics, and to determine the relationship between those attitudes and conceptions, as well as achievement. This study informs the practice of teaching statistics by giving insight to statistics instructors as to what student attitudes exist, as well as what conceptions and misconceptions students possess. Also, this study provides a catalyst for changing recognized attitudes and correcting misconceptions, which in turn will consequently lead to higher student achievement and a better overall understanding of statistics.
Janet C. Vassilev (UC Riverside)
Can you win the Hat
Game?
We will discuss how to play the Hat Game. Without knowing some discrete math, your chances of winning are not too high. The three person game gives a good starting point to discuss the probability of winning with students in the middle grades or higher. To win the n person game, we will learn the basics about vector spaces over Z2 and error correcting codes, in particular Hamming Codes, to win the game with a very high probability.
Martin Flashman (
Dynamic Visualization
of Calculus I-III
Professor Flashman will explain and use free graphing technology (Winplot) to illustrate how to visualize concepts related to functions and derivatives in calculus of one and several variables without graphs. The treatment is suitable for any introductory treatment of the concepts. Based on mapping (transformation) figures, this approach allows students to understand the concepts in an n-dimensional context without any change in presentation from that given for the ordinary derivative.
Points on
Hyperboloids and Units in an Integral Group Ring
There is a correspondence between
the group of units in Z[D4] and integer points on
hyperboloids of the form X2 +
Y2 = Z2 + n where n
is a positive integer of the form c(c-1). In this talk, we will discuss
methods for finding integer points on such hyperboloids and discuss the
relationship to units in the integral group ring of the dihedral group of order
8. This project was the work of third year undergraduate research student
Chester Weatherby at Mount Allison University
in the summer of 2003 building on projects done by Paul Moore in 1991 and
Elizabeth Jenkins in 1993, as well as research papers by Jespers,
Goodaire, Parmenter, Leal
on integral group rings. This talk should be accessible to undergraduate
students.
Resolutions of
modules over commutative rings
A vector space over the real
numbers is a set V equipped with an addition and scalar multiplication that
satisfies certain natural axioms. Given a ring R, an R-module is defined
similarly, except that the set of scalars is R instead of the real numbers. The
R-modules have become central objects of study in abstract algebra, but they
are usually less simple than vector spaces. For instance, an R-module in
general does not possess a basis. In some situations, this new level of
complexity can be demystified by "resolving" the module, a process
intimately related to the fundamental theorem of finitely generated abelian groups. In this talk I will give an overview of
these ideas, culminating in David Hilbert's famous Syzygy
Theorem, describing the structure of modules over a polyno
The Power of
Iteration
In elementary mathematics, the
technique of iteration has been applied to multiple
integration and to summation to good and, sometimes, surprising
advantage. In Analytic Number Theory, however, iteration is not just a useful
side show but serves as the backbone of the entire body of theory. The purpose
of this talk is to take a 'fresh' look at this very important technique in the
context of
Playing games using
Cryptography
Ever wonder how those online casinos work? We will go over some simple examples of how to play games over the internet using cryptography. The materials are taken from a short course that I gave to a group of high school students last year so the mathematics involved is very elementary.
WeBWorK
WeBWorK
is a computer package that serves up homework problems to students over the
web, and grades students' answers so they get immediate feedback. Problems are already written for several
courses: precalculus, three semesters of calculus,
statistics, some linear algebra, and statistics. It's possible to write your
own problems. WeBWorK was written by math faculty and
grad students at the
Toukaiddine Petit (
Strong rigidity of
Lie algebras
We call a finite-dimensional
complex Lie algebra L strongly rigid if its universal enveloping algebra U(L) is rigid as an associative algebra, i.e. every formal
associative deformation is equivalent to the trivial deformation. In quantum
group theory this phenomenon is well-known to be the case for all complex semisimple Lie algebras. We show that a strongly rigid Lie
algebra has to be rigid as Lie algebra, and that in addition its second scalar cohomology group has to vanish (which excludes nilpotent
Lie algebras of dimension greater or equal than two). Moreover, using Kontsevitch's theory of deformation quantization we show
that every polyno
The Mathematics of
Peg Solitaire
We will consider the game peg solitaire, more commonly known as Hi-Q. The instructions for the game claim that if you can end up with one peg in the center of the board, you are a perfect Hi-Q genius, but if you end up with one peg anywhere else on the board, you are merely outstanding. We will use the context of peg solitaire to introduce the concept of a group. We will then use two group homomorphisms and the Klein-4 group, to show that the Hi-Q instructions might be more accurate if they replaced the word "outstanding" by "oblivious." We will also investigate different versions of the game on different boards. Most importantly, there will be M&M's.
The Socratic Method
Adapted to the Mathematics Classroom
Zeno (apparently) invented and Socrates developed the dialectic method of teaching: that is, teaching through questioning. The Socratic method involves skepticism, discussion, a search for precise definitions of terms, and follows ideas to their logical conclusion through rigorous thinking and focused questioning. Can we - and should we - use this method in our classes today? If not in its entirety, can we use it partially? Which parts? When? How? In what types of classes? We will discuss these issues in the context of courses that many of us teach.
Alex Stanoyevitch (
Logistics of Air
Travel
In this talk, we will show how to model a network of airline routes using a directed graph and the so-called incidence matrix. Many interesting questions about the network can be answered by performing appropriate computations with this incidence matrix and other related matrices. We will show how to answer questions like: What is the most number of separate flights one would need in order to get from any one city in the network to a different city? If we call this number the worst case scenario number for the network, another useful question would be: If the manger of the network is interested in reducing the worst case scenario number, could this be done by adding a single new route to the network? If so, by how much, and which would be the best route(s) to add to accomplish this reduction? Same question with two new flights? Although some of the concepts that we touch upon will be rather sophisticated, the formal prerequisites for this talk will be quite minimal, and everyone will be able to learn some things from it.
Multiplicative
Lattices and Generalizations of Properties in Commutative Algebra
The set of ideals in a commutative ring is a partially ordered set ordered by inclusion. The basic properties satisfied by this structure of the set of ideals lead to the abstract idea of a multiplicative lattice. This more general concept can be used to obtain results that can then be applied to structures in other parts of mathematics. We will consider generalizing the notions of a principal ideal, a Noetherian ring, and the integral closure of a ring and look at current research in this area.
The Bargmann Transform and Windowed Fourier Localization
Operators which localize in both time and frequency are of interest for applications in signal analysis. I consider the Gabor-Daubechies windowed Fourier localization operators Lφw, with ``symbol" (or ``weight function") φ and ``window" w. There is an interesting connection between these operators and Berezin-Toeplitz operators, via the Bargmann isometry β. For ``window'' w a finite linear combination of Hermite functions and some interesting classes of ``symbols'' φ, L. A. Coburn conjectured an equivalence of the form
β Lφw β-1=C*Mφ C=T(I+D)φ,
where T(I+D)φ is a Berezin-Toeplitz
operator with symbol (I+D)φ, Mφ
is the operator of ``multiplication by φ,
C=C(w) is a precisely determined
operator, and D=D(w) is a constant-coefficient
linear differential operator with constant term 0. I settled Coburn's
conjecture affirmatively by obtaining the exact formulas for C and the linear differential operator D. Calculation for a simple window
function will be demonstrated, and the formulas of C and D for the
conjectured result outlined above will be discussed.
Bogdana Georgieva (
Calculus of
Variations - The Mathematics of Optimization
Everything we do, we want to do it as efficiently as possible. For example, a student strives to get the highest possible grades for her/his efforts, a company endeavors to maximize its profits, ... We all optimize our efforts sometimes consciously, sometimes subconsciously. No one is surprised by such observations. However, not everyone knows that, in a sense, Nature acts in the same way. Many of the fundamental laws of physics and geometry follow from a principle of minimization / maximization of a certain quantity. The branch of mathematics which studies this principle and its applications is called the Calculus of Variations. In this talk I will attempt to give you some idea of the problems which can be solved with the methods of the Calculus of Variations. I will also talk about some examples for applications to nonconservative processes, to the nonlinear damped Klein-Gordon equation, and to the propagation of electromagnetic waves in conductive medium.
Fairly recently, Gustav Herglotz formulated a variational principle which is more general than the classical variational principle and contains the classical variational principle as a special case. This variational principle is important for a number of reasons. Notably, it is closely related to contact transformations and it can give a variational description of nonconservative processes.
Vanishing of
functions on intersections of algebraic varieties
Let X,Y be subvarieties
of the affine space of dimension n
over a field k, and fix a polyno
Kurano and Roberts' work on Serre's Positivity Conjecture for intersection multiplicities provides one such bound as well as our motivation for investigating this question. We will describe a sharper, more symmetric bound that follows from our generalization of Serre's dimension inequality. We will introduce the relevant tools and ideas from commutative algebra used to understand this problem, illustrating each one with concrete examples.
Alex Kugushev (CyberGnostics, Inc.)
Presentation of CyberStats, a Web-driven Introduction to Statistics
CyberStats is a "living" electronic textbook. Students internalize statistical concepts by interacting with hundreds of simulations and calculations and immediate-feedback practice items. CyberStats is conceptual, not computational software (though it incorporates a computational component). It provides a learning opportunity that cannot be delivered in print and is equally effective for on-campus as for distance learning courses. The presentation will consist of an online demonstration of elements comprising CyberStats. The presenter will cover both instructional and learning elements. The learning elements will be presented by showing how a typical CyberStats Unit causes a student to interact with concepts and achieve mastery thereof. It will also cover tutorial aspects of CyberStats. The instructional elements will show the assistance available to the instructor: an integrated course management system, testing facilities, with a test bank and grade book allowing automatic test grading, full reports on all students' activities, a variety of communication devices, tips on how to teach successfully with the Web, and other useful items.
The Kontsevich Integral
The Kontsevich integral is an invariant of knots in 3-space. Sometimes it is also called universal finite type invariant because it contains the information of all finite type (Vassiliev) knot invariants. The Kontsevich integral is represented as an infinite series of terms whose coefficients are multidimensional integrals. In the simplest situation the first nontrivial coefficient is a very special double integral that can be understood by undergraduate students taking multivariable calculus. I will start the talk with this example and try to keep the level accessible for undergraduate students for as long as possible. At the end I am going to explain the Hopf algebra structure on chord diagrams which is highly important for finite type invariants.
Deborah Koslover (UC Irvine)
Bloch Electron in a
Perpendicular Magnetic Field
We study a model of an electron on a two dimensional crystal lattice subjected to a perpendicular magnetic field. We determine how the structure and spacing of the lattice as well as the strength of the magnetic field affect the motion and allowed energy levels of the electron.
Miriam Nuño (
A Mathematical Model
of Influenza: The Role of Cross-Immunity and Host-Isolation
Influenza virus infects 5% to 20%
of the
Freddy Van Oystaeyen
(
Introduction to
Non-commutative Geometry
Stefaan Cenepeel (Free University
The Brauer Group and Corings
Ioana Mihaila (Cal
Poly
The Art of
Multiplicative Periodic Functions on C\{0}
Periodic Functions are a common occurrence in mathematics, but on the punctured complex plane we can actually consider multiplicative periodic functions. Can these functions be explicitly constructed, and are they useful?
Jennifer Switkes (Cal Poly
On the Means of
Deterministic and Stochastic Populations
Familiar results for differential equation birth-death-immigration-emigration population models are compared with the expected population sizes predicted by related stochastic models. Although under standard assumptions the results are not equivalent, by removing certain restrictive modeling assumptions the two types of models can be reconciled.
Logic and Mathematics
On the Milnor
Conjecture
In 2002 V. Voevodsky was awarded the Fields Medal for his development of a new homotopy and cohomology theory for algebraic schemes; a consequence of this construction is the proof of the Milnor Conjecture, a problem that awaited a solution for twenty-six years. The aim of this talk is to give a description of the conjecture, explain how it works in a couple of particular cases, and give an outline of Voevodsky’s proof.
Patrick Callahan (
Mathematics Teachers: Supply and Demand in
Mathematics Teachers: Supply and
Demand in
Discovery Based
Learning: How and Why I teach Math without a Textbook
Discovery Based Learning is a teaching method that gets students to do research at their level. The objective of Discovery Based Learning is to give students experiences that parallel what research mathematicians do, which is prove theorems. In this talk, I will discuss how DBL is implemented for different math courses, with the emphasis on Advanced Analysis. I will also give a survey of some evidence that shows why DBL is appropriate, and give examples of how some of the essential elements can be used in almost any math course.
On the Classification
of Semisimple Hopf Algebras
Hopf algebras appear as invariants in various other parts of mathematics, such as topology (in knot theory) and mathematical physics (in conformal field theory). Thus the classification of Hopf algebras is of some interest outside algebra. At least for (finite-dimensional) semisimple Hopf algebras over the complex numbers, many of the basic results resemble classical facts about finite groups, with the order of the group being replaced by the dimension of the Hopf algebra. For example, the dimension of a Hopf subalgebra always divides the dimension of the Hopf algebra, the analog of Lagrange's theorem. We survey these results, and then discuss some cases when the Hopf algebras do not behave like groups.
Calculus and planimeters
According to the Merriam-Webster
dictionary, a planimeter is “an instrument for
measuring the area of a plane figure by tracing its boundary line”. Even
without knowing how a planimeter works, it is clear
from the definition that the idea behind it is that one can compute the area of
a figure just by “walking” on the boundary. For someone who has taken calculus,
this immediately suggests Green's Theorem. The aim of this talk is to clarify
why this principle works. We do this by using points of view from linear
algebra to elementary plane geometry in order to obtain an intuitive
justification for Green's Theorem. The talk is based on joint work with
Introduction to Mathematica
Mathematica is a very elaborate high level programming package/language with tools that enable users to simplify algebraic expressions, solve systems of equations, integrate, differentiate, solve systems of differential equations, create 2D and 3D graphics, create and play sounds, create and play animations, format documents using standard mathematical notations, write programs, and a whole host of other things. I got interested in it when I wanted to create different kinds of map projections using real geographic data (yes, it contains that data too). I will introduce you to some Mathematica basics and we will try out some things on workstations in the WH C155 computer lab. There are plenty of seats so everyone should be able to run Mathematica for themselves.
A Probabilistic Approach to Generation and Properties of Finite Groups
It was finally shown in 1984 that every finite simple group can be generated by a pair of elements. The initial proof of this result was obtained by exhibiting such a pair for every simple group. In the last 20 years, a new approach has been used -- looking at the size of the set of pairs of elements that do generate. We will also discuss related questions about determining properties of groups by considering subgroups generated by pairs of elements.
Arthur Benjamin (
Counting on Determinants
We demonstrate how determinants
solve many interesting combinatorial problems. Determinants count non-intersecting
lattice paths, spanning trees, and permutations with specified descent points.
Elegant proofs of these results are based on the definition of the determinant
and occasionally the principle of inclusion-exclusion. This talk is based on
joint work with Naiomi Cameron of
David E. Radford (
Knots and Algebras
There are many ways of associating numbers, or other algebraic objects, to knots in order to distinguish different knots. We will explore a very intuitive way of doing this based on a "bead sliding" algorithm due to Kauffman. Plenty of examples will be given to illustrate how the algorithm works.
Tom Love (CSUDH)
Hidden Symmetries and the Internal Structure of Elementary Particles
Some manifolds have hidden symmetries. Exploring those symmetries leads to a new mathematical model of elementary particles in terms of differential operators and the possibility of new conservation laws. Our results show that matrix methods are inadequate for the study of elementary particles.
Piotr Kowalski (
Algebraic Groups With An Extra Structure
This is joint work with Anand Pillay. We work over a field with some extra structure, e.g. a derivation. Then we consider a category expanding the category of algebraic varieties, where an extra structure on a variety is related to the extra structure on the field. These categories were used by Anand Pillay to obtain easy proofs of some diophantine results. We will show that structures coming from groups in these categories enjoy certain logical property (quantifier elimination).
More Fair Games
Let n ≥2 be an integer. The n-color matching game is a 2-person game in which the players, called M and N, each draw a ball at random from a bag containing balls of n different colors. M wins the game if the balls drawn are of the same color, otherwise N wins. A game is fair if both players have the same chance to win. We are interested in finding Fn, the set of n-color fair games and the set of numbers. In particular, we will give a description of F3. This is joint work with Jacqueline Barab.
A survey of Galois corings
We introduce Galois corings and give a survey of properties that have been obtained so far. The definition is motivated using descent theory, and we show that classical Galois theory, Hopf-Galois theory, and coalgebra Galois theory can be obtained as special cases.
Homeomorphisms of Compact Totally Disconnected Metric Spaces
We define the limit index i(x) of a point x in a compact metric space. (Roughly: Isolated points have index 0, limit points have index 1, limit points of limit points have index 2, and so on.) We then consider homeomorphisms f of a compact, totally disconnected metric space into itself in terms of in terms of how i(x) and i(f (x)) are related. This leads to a theorem that characterizes the uniform closure of the set of all homeomorphisms of a compact totally disconnected metric space into itself. The results to be described appear in Proc. AMS 84 (1982), 264-266. Prerequisites: basic facts about limit points, compactness, metric spaces, and uniform convergence.
Another Method on Computing MTBF for a k-out-n: G Repairable System
System
It is often necessary to calculate MTBF (mean time between failures) as quickly as possible in order to make timely design decisions. An important system for which such calculation must be made is a k-out-of n: G repairable system with unlimited repair and exponential interfailure and repair times at the unit level. Although a general formula is known, but it is not easily remembered nor derived. In this talk, we would like to present a new method for deriving MTBF in this situation that is easily reproduced quickly by remembering a few simple concepts.
Michael W. Leonard (UC
Limited-Memory Quasi-Newton Methods: Recent Developments
Problems from all areas of science and engineering can be posed as optimization problems. An optimization problem involves a set of independent variables, and often includes constraints or restrictions that define acceptable values of the variables. The solution of an optimization problem is a set of allowed values of the variables for which some objective function achieves its maximum or minimum value. Quasi-Newton methods for optimization define an approximate Hessian that incorporates curvature information accumulated over a number of iterations. Characterization of the subspace on which curvature is known leads to the definition of limited-memory reduced-Hessian quasi-Newton methods, that require less storage and work than their conventional counterparts. We will discuss the current theoretical developments in this area, and present the most recent numerical results.
03/12/03
Modules over Hypersurface Rings
It is well known that a finitely generated abelian group is a direct sum of copies of Z and Z/(pk), for positive integers k and prime numbers p. It is also well known that given a finite group G we may describe all the finitely generated modules over the group C-algebra C[G] using the so called representation theory. Here we present some descriptions of finitely generated modules over hypersurface rings of type K[[X_1,...,X_n]]/(f), where K is a field and f is a non-invertible, non-zero formal power series.
David Marker (
Algebraic Nonstandard Models of Exponentiation
We outline an algebraic construction of nonstandard models of the real numbers with exponentiation. These nonstandard models provide useful asymptotic information about real functions. We will show how they can be used to answer a question of Hardy's .
An (in)equality with a long mathematical
history
This inequality, or the equality associated to it, appears in the classification of Dynkin diagrams (and therefore in the classification of Coxeter graphs and complex semisimple Lie algebras), but also in groups acting on spheres and elementary plane geometry. No prerequisites are necessary.
The Match Game
Seldom has a problem in elementary mathematics led to more beautiful interconnections. The main result of analyzing this family of simple, probabilistic games is one pretty theorem and three different proofs.
The Mathematics of Sound and Tuning Musical Instruments
In this talk I will discuss the Mathematics behind tuning musical instruments by discussing equal temperament and Pythagorean tuning. I will also explain why one can tell the difference between different sounds, and why some notes sound better when played together than others. This talk is intended for a general audience, and no special mathematical or musical background is required.
Inferring Expectations with the Kalman
Filters
Behavior is frequently influenced or driven by expectations. In his talk, Rod would like to demonstrate how Kalman Filters can be used to infer expectations from observed behavior in Economics.
A Cute Application of Model Theory
The talk will be an invitation to
Model Theory. I will give a very brief introduction to the subject and give an
interesting proof to the following theorem of Ax (and later by A. Borel): If a polyno
Silvia Heubach (
Counting Compositions: Patterns and Combinatorial Proofs
A composition of n is an ordered sequence of positive integers whose sum is n. A palindromic composition or palindrome of n is a composition that reads the same from left to right as from right to left. We will give methods to create all compositions and palindromes of n and use these methods to derive properties and count characteristics of the compositions and palindromes (total number, number of rises, levels and drops, number of + signs). We will also count how often a particular integer k occurs among all the compositions and palindromes of n, respectively, and look at patterns among these values and their combinatorial proofs.
This talk should be accessible to undergraduates who have taken a course in Discrete Mathematics.